The Phragmén-Lindelöf Theorem

Abstract

The maximum principle for analytic functions of a complex variable, which states that the absolute value of an analytic function attains its maximum on the boundary, is derived from the fact that the function cannot have a maximum of its absolute value in any interior point. It is thus only proved for compact regions. And indeed it is not valid for noncompact regions, as the following example shows. We consider f(s) = e^{cos s} in the strip S: —π/2 ≦ σ ≦ π\2. It is obviously regular in the strip and on its boundaries. On the boundaries we have f(±π/2 + it) = e^±isinht = e^±isnht and thus |f(±π\2 + it) | = 1. However f(it) = e^{cosh t} ≧e^(1/2)exp|t| which tends to ∞ as |t| → ∞. This example is also instructive in so far as it shows the least order of growth that a function bounded on the boundary must have if it does not remain bounded in the interior of the strip.